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The problem involves sketching the graph of the piecewise function \( f(x) \), which is defined as follows:

\[
f(x) = \begin{cases} 
x + 1 & \text{if } x < -2, \\
-2x - 3 & \text{if } x \geq -2.
\end{cases}
\]

### Step 1: Analyze Each Piece of the Function
1. **For \( x < -2 \)**: The function is \( f(x) = x + 1 \), which is a linear equation with slope 1 and intercept 1.
2. **For \( x \geq -2 \)**: The function is \( f(x) = -2x - 3 \), which is also linear but has a slope of -2 and a y-intercept of -3.

### Step 2: Graphing Each Piece
- **First piece \( f(x) = x + 1 \) for \( x < -2 \)**:
   - This is a straight line. We can calculate a couple of points to plot:
     - When \( x = -3 \), \( f(-3) = -3 + 1 = -2 \).
     - When \( x = -4 \), \( f(-4) = -4 + 1 = -3 \).
   - Since this part is valid only for \( x < -2 \), it will not include \( x = -2 \), and we'll use an open circle at the boundary \( x = -2 \).

- **Second piece \( f(x) = -2x - 3 \) for \( x \geq -2 \)**:
   - This is also a straight line. We can calculate points for this part:
     - When \( x = -2 \), \( f(-2) = -2(-2) - 3 = 4 - 3 = 1 \).
     - When \( x = 0 \), \( f(0) = -2(0) - 3 = -3 \).
   - This line includes \( x = -2 \), so the point \( (-2, 1) \) will be a closed circle.

### Step 3: Domain in Interval Notation
Since the function is defined for all real values of \( x \), the domain is:

\[
\text{Domain} = (-\infty, \infty)
\]

Would you like me to graph this piecewise function for you? Or provide more details?

### Related Questions:
1. How do you determine the boundary points for piecewise functions?
2. What happens when a piecewise function has a discontinuity at a certain value of \( x \)?
3. How can you verify if a piecewise function is continuous at the boundary point?
4. How does the slope of a linear equation affect the steepness of the line?
5. Can a piecewise function have non-linear pieces (such as quadratic or exponential)?

### Tip:
When graphing piecewise functions, always pay attention to open and closed circles at the boundaries to avoid mistakes in representing which parts of the graph are included or excluded.

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. f(x) = {x + 1 if x < -2, -2x - 3 if x >= -2}

Learn how to sketch the graph of a piecewise function where f(x) = x + 1 for x < -2 and f(x) = -2x - 3 for x >= -2. Explore linear equations and how to analyze different sections of a piecewise function. Includes domain in interval notation.

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